Exercises: Concepts

Many things happen in our daily lives, but not all of them depend on chance. State which of the following situations are random experiments and which are deterministic:

1. a) Tossing a coin and observing whether it lands on heads or tails.
2. b) Calculating how much $3$ movie tickets will cost if each one is worth $\$8$.
3. c) Asking a classmate what their favorite color is.
4. d) Dropping a stone from a balcony and seeing if it falls downwards.
5. e) Drawing a card from a deck of cards without looking.

A video game company wants to know the favorite genre of the students at your school. To avoid asking all $800$ students, they decide to survey $10$ students from each class in the school.

Identify the following elements of the statistical study:

• Population:

• Sample:

• Individual:

Classify the following statistical variables as qualitative (if they express a quality or characteristic) or quantitative (if they are expressed by a number):

1. a) The name of the most used apps on your friends' cell phones.
2. b) The number of siblings each student in your class has.
3. c) The brand of sneakers worn by the players of a team.
4. d) The maximum temperature reached in your city each day this week.
5. e) The preferred sport of the members of an athletics club.

Within quantitative variables, we distinguish between discrete (when they take isolated values, usually integers) and continuous (when they can take any value in an interval, such as decimals). Classify these variables:

1. a) The number of goals a team scores in each league match.
2. b) The exact time it takes a swimmer to cover $50$ feet.
3. c) The amount of books in the students' backpacks.
4. d) The height of the members of the US national basketball team.

A group of friends is discussing which movie to go and see at the movie theater. They have noted down their preferences:

Adventure, Comedy, Horror, Adventure, Adventure, Horror, Comedy, Adventure, Comedy.

Answer the following questions about the basic concepts:

1. a) What is the statistical variable being studied?
2. b) What type of variable is it?
3. c) What is the absolute frequency (number of times it appears) of the "Adventure" category?

Sometimes it is impossible to study the whole population and that is why we use a sample. Explain in which of these cases you think it is mandatory to use a sample instead of asking the whole population:

1. a) We want to know the mean grade of the last math exam in your 6th grade class.
2. b) A light bulb factory wants to know how many hours they last before burning out.
3. c) The principal of the school wants to know how many students have come to class today.
4. d) A hospital wants to analyze the blood group of all the inhabitants of a state.

Look at the following bar chart representing the pets of the students in a class and answer based on the concepts learned:

1. a) What is the sample size (total number of surveyed students)?
2. b) What is the studied variable and what type is it?
3. c) What is the absolute frequency of the variable "Cat"?

Imagine you want to carry out a study on the reading habits of teenagers in your city. Give an example of:

1. A question whose answer is a qualitative variable.

2. A question whose answer is a discrete quantitative variable.

3. A question whose answer is a continuous quantitative variable.

A well-known streamer wants to know what the favorite video game of their followers in the US is. To do this, they carry out a quick poll during a live stream in which $1500$ people participate. Classify the following variables collected in the study as qualitative or quantitative:

• a) Title of the favorite video game.

• b) Number of hours they play a week.

• c) Console or platform used (PC, PS5, Switch, etc.).

• d) Amount of money spent on skins or battle passes a month.

• e) Rating of the live stream from 1 to 10.

To improve the quality of the food in the cafeteria of a large school with $1200$ students, the school council decides to ask the opinion of the first $5$ students from each class who enter the school on Monday morning. Identify the following elements of this statistical study:

• a) Population.

• b) Sample.

• c) Individual.

State whether the following phenomena are random or deterministic, briefly justifying your answer:

• a) Knowing if it will rain in your city exactly on your birthday next year.

• b) Calculating the price of $5$ movie tickets if you know that each one costs $\$8,50$.

• c) Predicting the number that will come up when rolling a 10-sided die in a role-playing game.

• d) Knowing the grade of a math exam if you have left all the questions blank.

• e) Drawing a card from a standard deck of cards and it being the ace of diamonds.

In a study on the reading habits of teenagers in a city, the statistical variable is defined as: "Number of books read over the summer".

• a) What type of variable is this?

• b) If the study is carried out on a sample of $200$ young people and the sum of all their read books is $450$, what does the value $2,25$ represent in this context?

• c) Make up three possible values that an individual could take for this variable.

Look at the following diagram representing a study on the health of trees in a nature reserve that has $5000$ specimens. The forestry team has analyzed $150$ of them.

• a) What is the population size?

• b) What is the sample size?

• c) If the researchers record whether the tree has a pest or not, what type of variable are they studying?

An online clothing store wants to know the satisfaction level of its customers (Very dissatisfied, Dissatisfied, Neutral, Satisfied, Very satisfied).

• a) Classify this variable. Is it quantitative or qualitative?

• b) Although words are used, numbers are sometimes assigned ($1$ to $5$) to calculate the mean. Do you think this makes the variable quantitative? Justify your answer.

• c) If the store sends the survey to $500$ customers who bought yesterday, who is the "individual" in this study?

A cell phone manufacturer wants to test the screen resistance of a new model. The factory produces $10,000$ phones a day. For the study, $50$ phones are selected and dropped repeatedly from a height of $1,5$ feet until the screen breaks, recording how many drops each one withstands.

• a) Explain why in this case it is strictly necessary to use a sample instead of studying the whole population.

• b) Identify the studied variable and classify it.

• c) What would happen to the production if the manufacturer decided not to use a sample and analyze every individual in the population?

Identify the population and the individual in each of the following statistical studies:

• 1. We want to know the mean weight of the $250$ oz loaves of bread produced by a bakery in one day.

• 2. We want to know the favorite subject of 6th grade students across the US.

• 3. A vet analyzes the type of diet of the cats living in an urban colony in their neighborhood.

• 4. A light bulb company wants to know how many hours their energy-saving bulbs last before burning out.

A famous influencer wants to know what American teenagers think about the new time limit on a social network. To do this, they post a poll on their Instagram stories for 24 hours. $15.400$ people participate.

1. a) Identify the population under study and the sample used.
2. b) Do you think the sample is representative of the population? Justify your answer by analyzing two possible factors that could invalidate the results (biases).
3. c) Propose a more reliable method to choose a sample of $500$ individuals that truly represents all teenagers in the United States.

Classify the following statistical variables as qualitative or quantitative. In the case of quantitative variables, reflect on whether it makes sense to perform arithmetic operations (such as the mean) with their values:

1. a) The ZIP code of the residence of the students at a school.
2. b) The serial number of the computers in the library.
3. c) The daily mean temperature recorded in the schoolyard during the month of January.
4. d) The jersey number of the players on a basketball team.
5. e) The time, expressed in seconds, that a group of swimmers takes to cross a $50$-foot pool.

Note: Think about whether the "number" acts as a label or as a real measurement.

We are carrying out an experiment on the growth of some tomato plants under different types of light. We have $200$ plants in total.

1. a) If we decide to analyze only the plants that are in the row closest to the window, are we studying the population or a sample?
2. b) Define the individual of this study.
3. c) We want to measure three variables: the height of the plant in $cm$, the color of the leaves (dark green, light green, or yellowish), and the number of tomatoes each plant produces. Classify each variable.
4. d) Is this a random or deterministic experiment? Justify your answer.

In a study on fruit consumption in a class of $30$ students, some data from the frequency table has been lost. We know that the relative frequency ($h_i$) of the students who eat "3 pieces a day" is exactly $0,2$ and that the absolute frequency ($f_i$) of those who eat "1 piece a day" is three times that of those who eat "none".

If there are $2$ students who do not eat any pieces of fruit a day, logically complete the following data:

1. a) How many students eat $3$ pieces of fruit a day?
2. b) How many students eat $1$ piece of fruit a day?
3. c) What is the absolute frequency of those who eat "2 pieces a day", knowing that all the previous options add up to the total of the class?

Imagine you want to carry out a statistical study to convince the principal of your school that classes should start 15 minutes later to improve performance.

1. a) Clearly define the population and a suitable sample that is not biased.
2. b) Design three different variables that you could measure: one qualitative, one quantitative that is a count (integers), and another quantitative that is a measurement (with decimals).
3. c) Indicate what data collection instrument you would use for each one (survey, direct observation, or measuring device).

Identify whether the following phenomena are random or deterministic, explaining why in each case. For the random ones, describe their sample space:

1. a) Dropping a stone from a balcony and timing how long it takes to reach the ground (assuming there is no wind and the height is fixed).
2. b) The number that will win the first prize in the lottery draw.
3. c) Drawing a card from a standard deck of cards and observing whether it is a "face card" (Jack, Queen, or King).
4. d) Mixing an acid with a base in a chemistry laboratory following an exact proportion to obtain a salt.
5. e) The exact time it will take for the next bus to arrive at the stop.

A survey has been carried out on the number of children per family in a neighborhood of $200$ families. It is known that $15\%$ of the families have $3$ or more children.

1. a) What is the absolute frequency of the event "having 3 or more children"?
2. b) If the angle in a pie chart for families with "1 child" is $144^\circ$, how many families have exactly one child?
3. c) Calculate the relative frequency of the group of families that have "1 child".

Analyze the following situation: A city council wants to know if citizens are happy with the new bike lane. To save costs, they decide to ask only the people riding on the bike lane on a Saturday morning.

1. a) Is the sample random?
2. b) What type of bias do you think this study will present? Will the results overestimate or underestimate the general satisfaction of the city?
3. c) Propose a way of choosing the sample that includes people who do not use a bicycle so that the study is objective.
4. d) If the variable is "Degree of satisfaction (None, Little, A lot)", what type of variable is it? Could the arithmetic mean of this data be calculated? Justify why.